Best way to compute $\langle a|B|a \rangle$ in Cirq, where a is a state obtained running circuit A. And B is a different Quantum Circuit

Question

I am implementing RQAOA in Cirq. After running regular QAOA to find an optimal state a (This I have done successfully).

I need to calculate $\langle a|Z_iZ_j|a\rangle$ for all $i,j$ in MyGraph.edges().

How should I go about using state a found with the QAOA circuit, to calculate the expectation value of a different circuit with that state?

Answer 1

If you've already simulated the final state $|a\rangle$, something like the following should work:

qubits = cirq.LineQubit.range(nqubits)

# qubit order in the observables must match the qubit order in the circuit used to generate |a>
qubit_map = dict(zip(qubits, range(nqubits))) 

for (i, j) in MyGraph.edges():
  # make the Z_i*Z_j observable
  ZiZj = cirq.Z(qubits[i]) * cirq.Z(qubits[j]) 
  # compute desired expectation
  expectation_ZiZj = ZiZj.expectation_from_state_vector(a, qubit_map=qubit_map)

Also in the expression $\langle a | B | a \rangle$, $B$ is generally not a quantum circuit, it needs to be an "observable" (Hermitian operator).